Nope.
Atlassian, in my opinion, have made a really bad choice of words when they rebranded the three main options to Cloud, Server and Data-Centre.
Your Server service is probably running on "the" Cloud, which is not the same as a Cloud service running on (Atlassian) Cloud.
Atlassian Cloud services almost all run on <something>.atlassian.net. (There's a handful of really old ones on different urls, but they're still Atlassian urls). As your url is not an Atlassian Cloud one, you are running Server (or DC) on some other service, and you'll need to talk to the owners of that (which might be your organisation)
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That is not an Atlassian url, you will need to contact the people who own that site.
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